The list of famous British Muslims is very long; Britain has a long history of close relationships with both Arab and non-Arab Muslim countries : Egypt, Pakistan, Bangladesh, Aden etc. So it's no  surprise that today 1.8 million Muslims are living in U.K.

We all know of Sadiq Khan, London's new mayor. Other celebrities include Iqbal Sacranie, the secretary-general of the Muslim Council of Britain; Zaki Badawi who, as chairman of the Muslim Law (Sharia) Council UK, has issued a statement condemning  terrorism as contrary to Islam. Waheed Alli, a famous businessman, has become a Lord (Baron Alli of Norbury in the London Borough of Croydon); Sarwar Ahmed, newspaper publisher (Asian Times) ; Sir Anwar Pervez, a businessman and Britain's richest Muslim ; Shami Ahmed, businessman (Joe Bloggs jeans); Lady Pola Uddin, Labour peer ; Naseem Hamed, boxer ; and so many others.

Wikipedia has issued a longer list : https://en.wikipedia.org/wiki/List_of_British_Muslims